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Interpreting Betas in Regression w/ transformations

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  • #49663

    Ryan
    Member

    Hello,
     
    I have read that you can convert unstandardized beta coefficients from data that has been natural log transformed into a “percent change” interpretation in linear regression (Flanders et al., 1992).  According to Flanders and colleagues, you can conclude that “a one percent increase in the independent variable changes (increases or decreases) the dependent variable by unstandardized coeff/100 ___units. Conversely, if you had natural log transformed the dependent variable and not the independent variables, then you would multiply the unstandardized coefficient by a hundred.
     
    My analysis, however, involved taking the common (not natural) log of both predictors and leaving the DV in its original scale.
     
    The (partial) unstandardized coefficient for X1 in multiple linear regression with two log transformed IVs is:
    b1 = (SPlogx1y) – (SPlogx1logx2)(SPlogx2y)        —————————————————————         (SSlogx1)(SSlogx2) – (SPlogx1logx2)Sqrd
    I suppose without any further transformations, the interpretation of this coefficient would be that a one unit increase in the common log(x1) results in a ___ change in y, after holding constant the effects of common log(x2). Or is this an incorrect interpretation?
    In linear regression, the residuals are assumed to be additive with mean zero, but after transforming the IVs, the model becomes multiplicative with a nonzero mean. I believe Flanders (1992) “figured out” an easy way to interpret this model (with natural log transformed variables).
    My questions are:
    1.       Can I use the same conversion for a common log transformation, and therefore make the same interpretation?
    2.       Are there other ways to, in a sense, “back transform” the beta coefficients to a value that is easily interpretable (e.g. one unit increase in X, leads to __ change in y).
    Also, I realize there are other ways (perhaps better in many circumstances) to deal with assumptions not being met in linear regression. However, I’ve chosen to use transformations and would like to stick to solving this interpretation issue.
    Any help would be greatly appreciated!
    Please correct me if I have made incorrect statements in this post.
    Thanks,
    Ryan
     
    Reference:
    Flanders W.D., DerSimonian R., and Freedman, D.S. (1992). Interpretation of Linear Regression Models That Include Transformations or Interaction Terms. Annals of Epidemiology, 2, 735-744.

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    #170024

    Ryan
    Member

    I received help to answer my question. Please disregard original email.

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